INDEX vector field

A proof for this statement is constructed in accordance with the fact that the latter inequality accounts for the sign of the coefficient in the polynomial P(/.) at kn T, which in turn is associated with the index of a steady-state point for the vector field (151) [60]. If this coefficient is positive at any point of the positive orthant R z, zt > 0, i = 1, 2,. . ., n, then the steady-state point is unique. 

A global property function is usually expressed as the expectation value of an operator or as the derivative of such an expectation value with respect to an internal or external parameter of the system. In the Born-Oppenheimer approximation, the electronic wave function depends parametrically upon the coordinates of the n nuclei, and therefore a set of the 3 -6 linearly independent nuclear coordinates constitutes the natural variables for such a choice of the potential function. However, the manifold M on which the gradient vector field is bound can be defined on a subset of 1R provided q < 3n-6, for example the intrinsic reaction coordinate (unstable manifold of a saddle point of index 1 of the Born-Oppenheimer energy hypersurface) or the reduced reaction coordinate. 

Thus far we have only considered one (boson) vector field, namely, the direct product field R Xn of creation and annihilation operators. The coefficients of the creation and annihilation operator pairs in fact also constitute vector fields this can be shown rigorously by construction, but the result can also be inferred. Consider that the Hamiltonian and the cluster operators are index free or scalar operators then the excitation operators, which form part of the said operators, must be contracted, in the sense of tensors, by the coefficients. But then we have the result that the coefficients themselves behave like tensors. This conclusion is not of immediate use, but will be important in the manipulation of the final equations (i.e., after the diagrams have contracted the excitation operators). Also, the sense of the words rank and irreducible rank as they have been used to describe components of the Hamiltonian is now clear they refer to the excitation operator (or, equivalently, the coefficient) part of the operator. 

The index of a closed curve C is an integer that measures the winding of the vector field on C. The index also provides information about any fixed points that might happen to lie inside the curve, as we ll see. 

As X moves counterclockwise around C, the angle 0 changes continuously since the vector field is smooth. Also, when x returns to its starting place, 0 returns to its original direction. Hence, over one circuit, 0 has changed by an integer multiple of 2 r. Let [0] be the net change in 0 over one circuit. Then the index of the closed curve C with respect to the vector field f is defined as... 

To compute the index, we do not need to know the vector field everywhere we only need to know it along C. The first two examples illustrate this point. 

Unusual fixed points) For each of the following systems, locate the fixed points and calculate the index. (Hint Draw a small closed curve C around the fixed point and examine the variation of the vector field on C.)... 

This was used to derive Eq. (15). A special case of equation (56) was previously used to classify the ways to build a box The Poincare index theorem was extended by Hopf to vector fields on arbitrary manifolds. For vector fields with m isolated hyperbolic critical points, the Poincare-Hopf index theorem is " ... 

For underground excavation engineering, the deformation or displacement of the surrounding rock mass is a direct index of the stability status of the surrounding rock mass. Therefore, in this section, the displacement and the vector field of the displacement of the surrounding rock mass are discussed and analyzed in different cases firstly. 

It is often necessary to calculate the index of a fixed point. For a linear vector field, the index can be obtained in terms of the eigenvalues of the linear operator. A nonlinear vector field can be shown to be... 

The proof of this theorem is obtained by linearizing the equation = H to = in a sufficiently small neighborhood of the fixed point The vector fields I —H, I —L are then shown to be homotopic and therefore to have the same index. Now, the rotation of any completely continuous linear vector field I —L on a small sphere around has been calculated in [23] as (—1) therefore this must also be the rotation of the field I —H. When A = 1 is an eigenvalue of Eq.(A.9), then linearization of H is not legitimate because is a double root of the equation = H . A small change in the operator H will either separate the double root to two neighboring simple roots or will annihilate it. More specifically, we may consider the equation —H(a) =0 (a is a parameter) as defining implicitly f(a). Then >1=1 is the exceptional case in which the implicit function theorem fails to apply. In the case of >i=l Eq.(A.9) can be written as... 

This mathematical theory provides a partition of the space which is analogous to the more familiar partition made in hydrology in river basins delimited by watersheds. It relies on the study of a local function F(r) called the potential function. The potential function carries the physical or chemical information e.g. the electron density, the ELF (see below), or even the electrostatic potential [56-58]. In the cases treated in the present book, the potential function is required to be defined at any point of a manifold which is either for molecules or the unit cell for periodic systems. Moreover the first and second derivatives with respect to the point coordinates must be defined for any point. Its gradient W(r) forms a vector field bounded on the manifold and determines two kinds of points on the one hand are the wandering points corresponding to W(r ) f 0. and on the other hand are the critical points for which VF(rc) = 0. A critical point is characterized by the index Ip, the number of positive eigenvalues of the second derivatives matrix (the Hessian matrix). There are four kinds of critical points in... 

The topology of the 7 vector field deserves a careful and detailed investigation. Its most interesting features are observed in the proximity of an SP at which the modulus 7 vanishes. An SP is classified in terms of topological index i [85, 86], and of a (rank, signature) label [15, 55-57, 87-90]. A continuous, open or closed, path of SPs is referred to as stagnation line (SL), consisting of either vortex points (index = + 1), or saddle points (index = —1). 

In Chapter 11 we discussed the fundamental properties of modes on optical waveguides. The vector fields of these modes are solutions of Maxwell s source-free equations or, equivalently, the homogeneous vector wave equations. However, we found in Chapter 12 that there are few known refractive-index profiles for which Maxwell s equations lead to exact solutions for the modal fields. Of these the step-profile is probably the only one of practical interest. Even for this relatively simple profile the derivation of the vector modal fields on a fiber is cumbersome. The objective of this chapter is to lay the foundations of an approximation method [1,2], which capitalizes on the small... 

A few comments on the layout of the book. Definitions or common phrases are marked in italic, these can be found in the index. Underline is used for emphasizing important points. Operators, vectors and matrices are denoted in bold, scalars in normal text. Although I have tried to keep the notation as consistent as possible, different branches in computational chemistry often use different symbols for the same quantity. In order to comply with common usage, I have elected sometimes to switch notation between chapters. The second derivative of the energy, for example, is called the force constant k in force field theory, the corresponding matrix is denoted F when discussing vibrations, and called the Hessian H for optimization purposes. 

Here, N is a diagonal matrix of effective refractive indexes of the modes of the s-th section, and p and q are the column vectors of mode field amplitudes (C) and q (C), respectively. From these equations it is easy to find the relations between the mode field amplitudes p(C) and q(C) at the position C and those at the position shifted by within the same waveguide section s ... 

It is known that the complete electromagnetic field in each slice s can be expressed in terms of Hertz electric and magnetic vectors rf and rf that are both parallel with the x coordinate axis. (It is a consequence of the invariance of the refractive index distribution of the slice with respect of the coordinates y and x). The field components in the slice s are given by the expressions... 

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